A new fractal surface is being developed for heat transfer by Crinkle, Inc. It is projected to revolutionize the home heating industry. As an agent of Industrial Espionage Ltd., you have stolen some preliminary experimental data. The data indicates that for a plate, \(0.25 \mathrm{~m}\) long, with air as the heat transfer medium flowing at \(5 \mathrm{~m} / \mathrm{s}\), the temperature profile in the air was measured to be:

\[T(x=L)=T_{s}-1.0 \times 10^{5} y+1.5 \times 10^{5} y^{3} \quad \text { plate surface at } y=0\]

a. What fractional improvement in the heat transfer coefficient (referenced to a conventional, smooth flat plate) was achieved by this new design?

b. Does the hydrodynamic boundary layer lie above or within the thermal boundary layer?